Theorem rexalim 2319

Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
rexalim	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Proof of Theorem rexalim

Step	Hyp	Ref	Expression
1		ralnex 2316	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
2	1	biimpi 113	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜑)
3	2	con2i 557	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-ral 2311  df-rex 2312

This theorem is referenced by: (None)